2540 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS
Then letting B be a monotone nonnegative, self adjoint operator, B : W →W ′ for W aseparable Hilbert space, consider the linear operator L : L2 (0,T,W )≡W → L2 (0,T,W ′)≡W ′ given as
Lu≡(
I− τh
h
)Bu.
Is it the case that L is monotone? Clearly it is linear and so it suffices to consider ⟨Lu,u⟩W ′,Wwhich equals
1h
∫ T
0⟨Bu(t) ,u(t)⟩dt− 1
h
∫ T
h⟨Bu(t−h) ,u(t)⟩dt
=1h
∫ T
0⟨Bu(t) ,u(t)⟩dt− 1
h
∫ T−h
0⟨Bu(t) ,u(t +h)⟩dt
≥ 1h
∫ T
0⟨Bu(t) ,u(t)⟩dt
−1h
∫ T−h
0
(12⟨Bu(t) ,u(t)⟩+ 1
2⟨Bu(t +h) ,u(t +h)⟩
)dt
=1
2h
∫ T−h
0⟨Bu(t) ,u(t)⟩dt +
1h
∫ T
T−h⟨Bu(t) ,u(t)⟩dt
− 12h
∫ T−h
0⟨Bu(t +h) ,u(t +h)⟩dt
=1
2h
∫ T−h
0⟨Bu(t) ,u(t)⟩dt +
1h
∫ T
T−h⟨Bu(t) ,u(t)⟩dt
− 12h
∫ T
h⟨Bu(t) ,u(t)⟩dt
=1
2h
∫ T−h
h⟨Bu(t) ,u(t)⟩dt +
12h
∫ h
0⟨Bu(t) ,u(t)⟩dt
+1h
∫ T
T−h⟨Bu(t) ,u(t)⟩dt− 1
2h
∫ T
h⟨Bu(t) ,u(t)⟩dt
=12h
∫ T−h
h⟨Bu(t) ,u(t)⟩dt +
1h
∫ T
T−h⟨Bu(t) ,u(t)⟩dt
− 12h
∫ T−h
h⟨Bu(t) ,u(t)⟩dt
+1
2h
∫ h
0⟨Bu(t) ,u(t)⟩dt− 1
2h
∫ T
T−h⟨Bu(t) ,u(t)⟩dt
=1
2h
∫ T
T−h⟨Bu(t) ,u(t)⟩dt +
12h
∫ h
0⟨Bu(t) ,u(t)⟩dt ≥ 0 (76.2.5)
The following is a restatement of Theorem 75.1.13